Question

f(x) = 5x^2
x = 10

find the derivative algebraically

Answer 1

@jim_thompson5910

Answer 2

What do you mean by that? Something like this? \[
f'(10)=\lim_{\Delta x->0} \frac{f(10+\Delta x)-f(10)}{\Delta x}\\
\ \ \ \ \ \ \ =\lim_{\Delta x->0} \frac{5(10+\Delta x)^2-500}{\Delta x}\\
\ \ \ \ \ \ \ =\lim_{\Delta x->0} \frac{5(100+20\Delta x+{\Delta x}^2)-500}{\Delta x}\\
\ \ \ \ \ \ \ =\lim_{\Delta x->0} \frac{500+100\Delta x+5{\Delta x}^2-500}{\Delta x}\\
\ \ \ \ \ \ \ =\lim_{\Delta x->0} \frac{100\Delta x+5{\Delta x}^2}{\Delta x}\\
\ \ \ \ \ \ \ =\lim_{\Delta x->0} [100+5{\Delta x}]\\
\ \ \ \ \ \ \ =100
\]

Answer 3

The derivative of f(x) using the limit definition is

\[\Large \lim_{h\to0} \frac{f(x+h)-f(x)}{h} \]

Answer 4

So we need to find f(x+h) to go forward

Answer 5

f(x) = 5x^2

f(x+h) = 5(x+h)^2 ... replace all copies of 'x' with 'x+h'

Then simplify/expand

f(x+h) = 5(x+h)(x+h)

f(x+h) = 5[x(x+h) + h(x+h)]

f(x+h) = 5[x^2+xh + hx+h^2]

f(x+h) = 5[x^2+ 2xh+h^2]

f(x+h) = 5x^2+ 10xh+5h^2

Answer 6

\[\lim x->0 5(x ^{2} +2hx + h ^{2}) -5h ^{2} / h \]